Invariant co-ordinate selection

David E. Tyler, Frank Critchley, Lutz Dümbgen, Hannu Oja

Research output: Contribution to journalArticlepeer-review

78 Scopus citations


A general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented. The method is based on the eigenvalue-eigenvector decomposition of one scatter matrix relative to another. In particular, it is shown that the eigenvectors can be used to generate an affine invariant co-ordinate system for the multivariate data. Consequently, we view this method as a method for invariant co-ordinate selection. By plotting the data with respect to this new invariant co-ordinate system, various data structures can be revealed. For example, under certain independent components models, it is shown that the invariant co- ordinates correspond to the independent components. Another example pertains to mixtures of elliptical distributions. In this case, it is shown that a subset of the invariant co-ordinates corresponds to Fisher's linear discriminant subspace, even though the class identifications of the data points are unknown. Some illustrative examples are given.

Original languageEnglish (US)
Pages (from-to)549-592
Number of pages44
JournalJournal of the Royal Statistical Society. Series B: Statistical Methodology
Issue number3
StatePublished - Jun 2009

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Affine invariance
  • Cluster analysis
  • Independent components analysis
  • Mixture models
  • Multivariate diagnostics
  • Multivariate scatter
  • Principal components
  • Projection pursuit
  • Robust statistics

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